The class of nilpotent groups is closed under subgroups and quotients, but not extensions

Prove that subgroups and quotient groups of nilpotent groups are nilpotent. (Your proof should work for infinite groups.) Give an explicit example of a group G which possesses a normal subgroup H such that H and G/H are nilpotent but G is not nilpotent.

We showed that subgroups and quotients of nilpotent groups are nilpotent in the lemmas to this previous exercise.

Consider now D_6. \langle r \rangle \leq D_6 is a normal subgroup of order 3, and thus \langle r \rangle \cong Z_3 is abelian, hence nilpotent. Moreover, D_6/\langle r \rangle \cong Z_2 is abelian, hence nilpotent. However, D_6 has distinct Sylow 2-subgroups \langle s \rangle and \langle sr \rangle (for example). By Theorem 3, D_6 is not nilpotent.

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